<h2>Problem 298</h2>
<div style="color:#666;font-size:80%;">25 June 2010</div><br />
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<p>Larry and Robin play a memory game involving of a sequence of random numbers between 1 and 10, inclusive, that are called out one at a time. Each player can remember up to 5 previous numbers. When the called number is in a player's memory, that player is awarded a point. If it's not, the player adds the called number to his memory, removing another number if his memory is full.</p>

<p>Both players start with empty memories. Both players always add new missed numbers to their memory but use a different strategy in deciding which number to remove:<br />
Larry's strategy is to remove the number that hasn't been called in the longest time.<br />
Robin's strategy is to remove the number that's been in the memory the longest time.</p>

<p>Example game:<br />
<center><table class="p298">
<tr>
  <th>Turn</th>
  <th>Called<br />number</th>
  <th style="text-align:right">Larry's<br />memory</th>
  <th>Larry's<br />score</th>
  <th style="text-align:right">Robin's<br />memory</th>
  <th>Robin's<br />score</th>
</tr>
<tr>
  <td>1</td>
  <td>1</td>
  <td style="text-align:right">1</td>
  <td>0</td>
  <td style="text-align:right">1</td>
  <td>0</td>
</tr>
<tr>
  <td>2</td>
  <td>2</td>
  <td style="text-align:right">1,2</td>
  <td>0</td>
  <td style="text-align:right">1,2</td>
  <td>0</td>
</tr>
<tr>
  <td>3</td>
  <td>4</td>
  <td style="text-align:right">1,2,4</td>
  <td>0</td>
  <td style="text-align:right">1,2,4</td>
  <td>0</td>
</tr>
<tr>
  <td>4</td>
  <td>6</td>
  <td style="text-align:right">1,2,4,6</td>
  <td>0</td>
  <td style="text-align:right">1,2,4,6</td>
  <td>0</td>
</tr>
<tr>
  <td>5</td>
  <td>1</td>
  <td style="text-align:right">1,2,4,6</td>
  <td>1</td>
  <td style="text-align:right">1,2,4,6</td>
  <td>1</td>
</tr>
<tr>
  <td>6</td>
  <td>8</td>
  <td style="text-align:right">1,2,4,6,8</td>
  <td>1</td>
  <td style="text-align:right">1,2,4,6,8</td>
  <td>1</td>
</tr>
<tr>
  <td>7</td>
  <td>10</td>
  <td style="text-align:right">1,4,6,8,10</td>
  <td>1</td>
  <td style="text-align:right">2,4,6,8,10</td>
  <td>1</td>
</tr>
<tr>
  <td>8</td>
  <td>2</td>
  <td style="text-align:right">1,2,6,8,10</td>
  <td>1</td>
  <td style="text-align:right">2,4,6,8,10</td>
  <td>2</td>
</tr>
<tr>
  <td>9</td>
  <td>4</td>
  <td style="text-align:right">1,2,4,8,10</td>
  <td>1</td>
  <td style="text-align:right">2,4,6,8,10</td>
  <td>3</td>
</tr>
<tr>
  <td>10</td>
  <td>1</td>
  <td style="text-align:right">1,2,4,8,10</td>
  <td>2</td>
  <td style="text-align:right">1,4,6,8,10</td>
  <td>3</td>
</tr></table></center></p>

<p>Denoting Larry's score by <var>L</var> and Robin's score by <var>R</var>, what is the expected value of |<var>L</var>-<var>R</var>| after 50 turns? Give your answer rounded to eight decimal places using the format x.xxxxxxxx .</p>
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